Dummit and Foote Section 1.3 Symmetric Groups - Exercise 10 states:

Prove that if is the m-cycle ( ... ) then for all i {1,2, ... m} we have ( ) = where k+i is replaced by its least positive residue mod m. Deduce that | | = m.

Can anyone help with this problem? Would be grateful for help!

I think I can see how this works but I am struggling with how to write a clear and valid formal proof.

Peter